First of all I wonder whether the British headdress in India was somehow designed to resemble the headdress of the Indo-Greeks and Greco-Bactrians? For example, this is the image of Eucratides I, the Greco-Bactrian king minted on a coin (Indo-Greeks wore similar helms):

To me it bears resemblance to a typical British helm of the time:

I also read a point of view that the British monarch was declared emperor of India because the Indo-Greek kings called themselves "Basileos of India". I wonder whether it is true and whether there were other parallels?